Learning about emulsions or foam

I am on a strict medical ketogenic diet which calls for high amounts of fat, and I am having trouble making sauces that stand up, instead having to drink my fat or have oil running all over the bowl I have to drink with a spoon. The fat to protein+carb ratio is very high, example below. Also, carbs are strictly limited so carb thickeners are pretty much out, e.g. starches, whereas thick oils like coconut oil are definitely in. I also don't eat soy b/c of the genetic modification. Can someone help me understand how I might use small amounts of egg yolks, or other ingredients to make the following example sauce palatable and thick enough to eat rather than drink?

15 gr coconut oil
14 gr MCT oil
10 gr coconut milk.
This is served over a 45-gr lamb burger, so the largest element by volume of this meal is the sauce. It is imperative that all the sauce is consumed to maintain the proper ratio, blood sugar and blood ketones.

The chef who supposedly designed this recipe gave directions to "serve with the blended oil and coconut milk." but this is very thin, watery if heated, and if not heated it's a lumpy pasty unappealing-looking cold dipping agent.
Thanks so much for any ideas and tips.